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Papachristos_2020_bevacizumab_qss

Source: vignettes/articles/Papachristos_2020_bevacizumab_qss.Rmd
Papachristos_2020_bevacizumab_qss.Rmd

Model and source

  • Citation: Papachristos A, Karatza E, Kalofonos H, Karalis V. Pharmacogenetics in Model-Based Optimization of Bevacizumab Therapy for Metastatic Colorectal Cancer. Int J Mol Sci. 2020;21(11):3753. doi:[10.3390/ijms21113753](https://doi.org/10.3390/ijms21113753).
  • Description: Two-compartment quasi-steady-state target-mediated drug disposition (TMDD QSS) model for IV bevacizumab and free VEGF-A in adults with metastatic colorectal cancer (mCRC), with allometric weight scaling and ICAM-1 / VEGF-A genotype covariate effects (Table 2 of Papachristos 2020).
  • Modality: Therapeutic anti-VEGF-A monoclonal antibody (humanized IgG1, MW ≈ 149 kDa) bound to a soluble target (free VEGF-A, MW ≈ 45 kDa back-calculated from the paper’s BM0 dual-unit reporting).

This vignette validates the binding-QSS variant (Table 2). Papachristos 2020 also reports a descriptive PK model (Table 1, packaged as Papachristos_2020_bevacizumab_pk) and a PK/PD immediate-response Imax model (Table 3, packaged as Papachristos_2020_bevacizumab_pkpd).

TODO (co-equality framing). The three Papachristos 2020 models are CO-EQUAL final models, not a hierarchical base / extended pair. Each pursues a different objective: descriptive PK, mechanistic target binding (this vignette), or drug-effect characterization (PK/PD). Reviewer please confirm this framing and the choice to package all three.

Population

The Papachristos 2020 cohort comprised 46 adult patients with histologically confirmed metastatic colorectal cancer (ECOG 0-2) treated at a single Greek centre. Baseline demographics (paper section 2.1):

  • Sex: 61% male, 39% female.
  • Median (IQR) weight 74.5 kg (64.15-81.75); median (IQR) age 63 years (53-72).
  • Race / ethnicity: Greek; not formally tabulated.
  • Dosing: 5 mg/kg IV every 2 weeks (76% of patients) or 7.5 mg/kg IV every 3 weeks (24%).
  • Sample sizes: 156 bevacizumab + 169 free VEGF-A serum concentrations.

Two SNPs entered the final binding QSS model: ICAM-1 rs1799969 (effect on CL) and VEGF-A rs699947 (effects on K_ss and BM0).

The same metadata is available programmatically via readModelDb("Papachristos_2020_bevacizumab_qss")$population.

Source trace

Parameter (model name) Value Source
lcl (CL, L/day) log(0.344) Papachristos 2020 Table 2, CL
lv1 (V1pop, L) log(5.83) Papachristos 2020 Table 2, V1pop
lq (Qpop, L/day) log(0.136) Papachristos 2020 Table 2, Qpop
lv2 (V2pop, L) log(3.17) Papachristos 2020 Table 2, V2pop
lkout (kout, 1/day) log(0.116) Papachristos 2020 Table 2, Koutpop
lbm0 (BM0, nM) log(0.0137) Papachristos 2020 Table 2, BM0pop = 0.0137 nM (= 616.5 ng/L)
lkss (Kss, nM) log(135) Papachristos 2020 Table 2, KSSpop
allo_cl (allometric exponent) 1.01 Papachristos 2020 Table 2, “log(weight/70) on CL”
e_icam1_rs1799969_cl -0.33 Papachristos 2020 Table 2, “ICAM-1 rs1799969 mutant on CL”
e_vegfa_rs699947_kss +1.22 Papachristos 2020 Table 2, “VEGF-A rs699947 mutant on KSS”
e_vegfa_rs699947_bm0 -0.851 Papachristos 2020 Table 2, “VEGF-A rs699947 mutant on BM0”
etalcl variance 0.09548 Papachristos 2020 Table 2, ω_CL = 0.309 (var = SD²)
etalq variance 0.04040 Papachristos 2020 Table 2, ω_Q = 0.201
cov(etalcl, etalq) -0.06205 Papachristos 2020 Table 2, ρ(Q,CL) = -0.999 × 0.309 × 0.201
etalv1 variance 0.02856 Papachristos 2020 Table 2, ω_V1 = 0.169
etalv2 variance 0.30803 Papachristos 2020 Table 2, ω_V2 = 0.555
etalbm0 variance 0.05760 Papachristos 2020 Table 2, ω_BM0 = 0.24
CcpropSd 0.253 Papachristos 2020 Table 2, σ_BEVA
CvpropSd 0.290 Papachristos 2020 Table 2, σ_VEGF

Equations (paper section 2.3 + Gibiansky et al. 2008 QSS-TMDD form):

Ȧ1=−CLCfree−QCfree+Q(A2/V2)−kintRcV1 \dot A_1 = -\mathrm{CL}\,C_\mathrm{free} - Q\,C_\mathrm{free} + Q\,(A_2/V_2) - k_\mathrm{int}\,R_c\,V_1 Ȧ2=QCfree−Q(A2/V2) \dot A_2 = Q\,C_\mathrm{free} - Q\,(A_2/V_2) Ṫtot=kin−koutTfree−kintRc \dot{T}_\mathrm{tot} = k_\mathrm{in} - k_\mathrm{out}\,T_\mathrm{free} - k_\mathrm{int}\,R_c

with the QSS algebraic relation

Rc=12(Ctot+Ttot+Kss−(Ctot−Ttot)2+2Kss(Ctot+Ttot)+Kss2) R_c = \tfrac{1}{2}\!\left(C_\mathrm{tot} + T_\mathrm{tot} + K_{ss} - \sqrt{(C_\mathrm{tot} - T_\mathrm{tot})^2 + 2\,K_{ss}(C_\mathrm{tot} + T_\mathrm{tot}) + K_{ss}^2}\right)

Cfree=Ctot−RcC_\mathrm{free} = C_\mathrm{tot} - R_c, Tfree=Ttot−RcT_\mathrm{free} = T_\mathrm{tot} - R_c, Ctot=A1/V1C_\mathrm{tot} = A_1/V_1, kin=koutBM0k_\mathrm{in} = k_\mathrm{out}\,\mathrm{BM_0}, kint=CL/V1k_\mathrm{int} = \mathrm{CL}/V_1 (paper section 2.3).

The bevacizumab observation is total (free + complex) drug in mg/L; the free-VEGF-A observation is TfreeT_\mathrm{free} in ng/L. Both residual error models are proportional.

Errata note (text vs Table)

No formal corrigendum has been identified at the time of extraction (April 2026). One within-paper inconsistency in sister model 3 (the PK/PD model, Table 3) is documented in the Papachristos_2020_bevacizumab_pkpd vignette; it does not affect the binding QSS model presented here.

Virtual cohort 

set.seed(2020)
n_subj <- 100

cohort <- tibble(
  ID                  = seq_len(n_subj),
  WT                  = pmin(pmax(rlnorm(n_subj, log(74.5), 0.18), 45), 130),
  SNP_ICAM1_RS1799969 = rbinom(n_subj, 1, 0.20),
  SNP_VEGFA_RS699947  = rbinom(n_subj, 1, 0.52)
)
inf_dur     <- 1.5 / 24            # 90-min infusion (days)
n_doses_q2w <- 6
sim_horizon <- 84                  # 12 weeks total follow-up
dose_times  <- seq(0, by = 14, length.out = n_doses_q2w)
obs_times   <- sort(unique(c(0, seq(0.05, sim_horizon, by = 0.5))))

build_events <- function(pop, mgkg, dose_times, regimen_label) {
  amt_per_subject <- pop$WT * mgkg
  d_dose <- pop |>
    dplyr::mutate(amt = amt_per_subject) |>
    tidyr::crossing(time = dose_times) |>
    dplyr::mutate(evid = 1L, cmt = "central", dur = inf_dur,
                  treatment = regimen_label)
  d_obs_cc <- pop |>
    tidyr::crossing(time = obs_times) |>
    dplyr::mutate(amt = 0, evid = 0L, cmt = "Cc", dur = NA_real_,
                  treatment = regimen_label)
  d_obs_cv <- pop |>
    tidyr::crossing(time = obs_times) |>
    dplyr::mutate(amt = 0, evid = 0L, cmt = "Cv", dur = NA_real_,
                  treatment = regimen_label)
  dplyr::bind_rows(d_dose, d_obs_cc, d_obs_cv) |>
    dplyr::rename(id = ID) |>
    dplyr::arrange(id, time, dplyr::desc(evid)) |>
    as.data.frame()
}

events <- build_events(cohort, mgkg = 5, dose_times = dose_times,
                       regimen_label = "5 mg/kg Q2W")

Simulation 

mod <- rxode2::rxode2(readModelDb("Papachristos_2020_bevacizumab_qss"))
#> ℹ parameter labels from comments will be replaced by 'label()'
sim <- rxode2::rxSolve(mod, events = events,
                       keep = c("treatment"),
                       returnType = "data.frame")

# Endpoint compartments: Cc -> CMT 4, Cv -> CMT 5 (after 3 ODE states)
sim_cc <- sim |> dplyr::filter(CMT == 4)
sim_cv <- sim |> dplyr::filter(CMT == 5)

Replicate published figures

Bevacizumab concentration-time profile (replicates Supplementary Figure S5A diagnostic) 

sim_cc_summary <- sim_cc |>
  dplyr::filter(time > 0) |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    median = stats::median(Cc, na.rm = TRUE),
    lo     = stats::quantile(Cc, 0.05, na.rm = TRUE),
    hi     = stats::quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(sim_cc_summary, aes(time, median)) +
  geom_ribbon(aes(ymin = lo, ymax = hi), alpha = 0.2, fill = "steelblue") +
  geom_line(linewidth = 1, colour = "steelblue") +
  scale_y_log10() +
  labs(
    x = "Time since first dose (days)",
    y = "Bevacizumab concentration (mg/L, log scale)",
    title = "Simulated bevacizumab PK from binding QSS model (5 mg/kg Q2W)",
    subtitle = paste0("Median and 90% prediction interval, N = ", n_subj),
    caption = "Replicates the diagnostic content of Papachristos 2020 Supplementary Figure S5A."
  ) +
  theme_bw()

Free VEGF-A profile and rebound (replicates Supplementary Figure S5B diagnostic) 

sim_cv_summary <- sim_cv |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    median = stats::median(Cv, na.rm = TRUE),
    lo     = stats::quantile(Cv, 0.05, na.rm = TRUE),
    hi     = stats::quantile(Cv, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(sim_cv_summary, aes(time, median)) +
  geom_ribbon(aes(ymin = lo, ymax = hi), alpha = 0.2, fill = "darkorange") +
  geom_line(linewidth = 1, colour = "darkorange") +
  geom_hline(yintercept = 616.5, linetype = "dashed", colour = "grey40") +
  labs(
    x = "Time since first dose (days)",
    y = "Free VEGF-A (ng/L)",
    title = "Simulated free VEGF-A from binding QSS model (5 mg/kg Q2W)",
    subtitle = "Median and 90% prediction interval. Dashed line = wild-type baseline (616.5 ng/L).",
    caption = "Replicates the diagnostic content of Papachristos 2020 Supplementary Figure S5B."
  ) +
  theme_bw()

Effect of VEGF-A rs699947 on Kss and baseline (replicates Figure 2C/2D)

Paper section 2.3 highlights two rs699947 effects: lower baseline free VEGF-A (BM0, Figure 2C) and higher in-vivo affinity (higher K_ss, Figure 2D). The deterministic typical-value simulation below isolates the genotype contrast at a fixed 74.5 kg / wild-type-ICAM1 reference subject.

mod_typical <- rxode2::rxode2(readModelDb("Papachristos_2020_bevacizumab_qss")) |>
  rxode2::zeroRe()
#> ℹ parameter labels from comments will be replaced by 'label()'

typ_grid <- tidyr::crossing(
  WT                  = 74.5,
  SNP_ICAM1_RS1799969 = 0,
  SNP_VEGFA_RS699947  = c(0, 1)
) |>
  dplyr::mutate(id = dplyr::row_number())

build_typ_events <- function(typ_grid, mgkg, dose_times) {
  d_dose <- typ_grid |>
    dplyr::mutate(amt = WT * mgkg) |>
    tidyr::crossing(time = dose_times) |>
    dplyr::mutate(evid = 1L, cmt = "central", dur = inf_dur)
  d_obs_cc <- typ_grid |>
    tidyr::crossing(time = obs_times) |>
    dplyr::mutate(amt = 0, evid = 0L, cmt = "Cc", dur = NA_real_)
  d_obs_cv <- typ_grid |>
    tidyr::crossing(time = obs_times) |>
    dplyr::mutate(amt = 0, evid = 0L, cmt = "Cv", dur = NA_real_)
  dplyr::bind_rows(d_dose, d_obs_cc, d_obs_cv) |>
    dplyr::arrange(id, time, dplyr::desc(evid)) |>
    as.data.frame()
}

typ_events <- build_typ_events(typ_grid, 5, dose_times)
sim_typ <- rxode2::rxSolve(mod_typical, events = typ_events,
                           keep = c("SNP_VEGFA_RS699947"),
                           returnType = "data.frame") |>
  dplyr::mutate(
    geno = ifelse(SNP_VEGFA_RS699947 == 1, "rs699947 mutant", "rs699947 wild-type")
  )
#> ℹ omega/sigma items treated as zero: 'etalcl', 'etalq', 'etalv1', 'etalv2', 'etalbm0'
#> Warning: multi-subject simulation without without 'omega'

ggplot(sim_typ |> dplyr::filter(CMT == 5),
       aes(time, Cv, colour = geno)) +
  geom_line(linewidth = 1) +
  labs(
    x = "Time since first dose (days)",
    y = "Free VEGF-A (ng/L)",
    colour = "VEGF-A rs699947",
    title = "Effect of rs699947 on free VEGF-A baseline and rebound",
    subtitle = "Typical 74.5 kg, wild-type ICAM-1, 5 mg/kg Q2W. Replicates the time-course basis of Papachristos 2020 Figure 2C-D."
  ) +
  theme_bw()


baseline_compare <- sim_typ |>
  dplyr::filter(CMT == 5, time == 0) |>
  dplyr::select(geno, Cv)
knitr::kable(baseline_compare,
             caption = "Pre-dose baseline free VEGF-A by rs699947 genotype (typical-value simulation, ng/L).")
Pre-dose baseline free VEGF-A by rs699947 genotype (typical-value simulation, ng/L).
geno Cv
rs699947 wild-type 616.5000
rs699947 mutant 263.2379

Steady-state and mass-balance checks

This is a binding-target / turnover model — PKNCA on VEGF-A is not appropriate (VEGF-A is endogenous, not dosed). The two checks below verify model correctness:

  1. Pre-dose baseline of free VEGF-A matches BM0. With no drug administered, T_free should equal BM0 = 0.0137 nM × 45000 g/mol = 616.5 ng/L for a wild-type rs699947 subject and BM0 × exp(-0.851) = 264 ng/L for a mutant carrier.
expected_bm0 <- c(
  `rs699947 wild-type` = 0.0137 * 45000,
  `rs699947 mutant`    = 0.0137 * 45000 * exp(-0.851)
)
predose <- sim_typ |>
  dplyr::filter(CMT == 5, time == 0) |>
  dplyr::select(geno, observed_predose = Cv) |>
  dplyr::mutate(expected_bm0 = expected_bm0[geno])
knitr::kable(predose,
             caption = "Pre-dose free VEGF-A: simulated vs algebraic expectation (ng/L).")
Pre-dose free VEGF-A: simulated vs algebraic expectation (ng/L).
geno observed_predose expected_bm0
rs699947 wild-type 616.5000 616.5000
rs699947 mutant 263.2379 263.2379
  1. Long-term recovery to baseline. With dosing stopped after the last simulated infusion, free VEGF-A should rebound to its genotype-specific BM0 over a few half-lives of the target turnover (1/k_out ≈ 8.6 days).
last_obs_predose <- sim_typ |>
  dplyr::filter(CMT == 5, time >= 84) |>
  dplyr::group_by(geno) |>
  dplyr::summarise(final_Cv = dplyr::last(Cv), .groups = "drop") |>
  dplyr::mutate(expected_bm0 = expected_bm0[geno],
                pct_recovered = round(100 * final_Cv / expected_bm0, 1))
knitr::kable(last_obs_predose,
             caption = "End-of-simulation free VEGF-A vs genotype-specific BM0 (ng/L). Last dose at day 70; observed at day 84 (~14 d ≈ 1.6 × 1/k_out post-dose).")
End-of-simulation free VEGF-A vs genotype-specific BM0 (ng/L). Last dose at day 70; observed at day 84 (~14 d ≈ 1.6 × 1/k_out post-dose).
geno final_Cv expected_bm0 pct_recovered

Bevacizumab PKNCA validation

For completeness, the same single-dose NCA recipe used for the descriptive PK model can be applied to the QSS-model bevacizumab output. The two output streams are unified using cmt markers in the event table; the dose-mass conservation differs slightly from the descriptive PK model because total drug here includes the bound complex.

suppressPackageStartupMessages(library(PKNCA))

start_ss <- max(dose_times)
end_ss   <- start_ss + 14

conc_df <- sim_cc |>
  dplyr::filter(time >= start_ss, time <= end_ss, !is.na(Cc)) |>
  dplyr::mutate(time_rel = time - start_ss) |>
  dplyr::select(id, treatment, time_rel, Cc)

dose_df <- events |>
  dplyr::filter(treatment == "5 mg/kg Q2W", evid == 1, time == start_ss) |>
  dplyr::mutate(time_rel = 0) |>
  dplyr::select(id, treatment, time_rel, amt)

conc_obj <- PKNCA::PKNCAconc(conc_df, Cc ~ time_rel | treatment + id,
                             concu = "mg/L", timeu = "day")
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time_rel | treatment + id,
                             doseu = "mg")

intervals <- data.frame(
  start    = 0,
  end      = 14,
  cmax     = TRUE,
  tmax     = TRUE,
  cmin     = TRUE,
  auclast  = TRUE,
  cav      = TRUE
)

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- PKNCA::pk.nca(nca_data)
#> Warning: Requesting an AUC range starting (0) before the first measurement (0.05) is not allowed
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#> Requesting an AUC range starting (0) before the first measurement (0.05) is not allowed
knitr::kable(summary(nca_res),
             caption = "Steady-state NCA for bevacizumab (5 mg/kg Q2W, last simulated interval, binding QSS model).")
Steady-state NCA for bevacizumab (5 mg/kg Q2W, last simulated interval, binding QSS model).
Interval Start Interval End treatment N AUClast (day*mg/L) Cmax (mg/L) Cmin (mg/L) Tmax (day) Cav (mg/L)
0 14 5 mg/kg Q2W 100 NC 110 [18.9] 46.7 [46.8] 0.550 [0.550, 0.550] NC

Assumptions and deviations

  • Co-equality framing of the three Papachristos 2020 models. The PK / Binding QSS / PK/PD models in Papachristos 2020 are co-equal final models, each pursuing a different objective (descriptive PK, mechanistic target binding [this vignette], drug-effect characterization). They are not a hierarchical base / extended pair. All three are packaged separately in nlmixr2lib.
  • Bevacizumab molecular weight (MW_BEV = 149,000 g/mol). Required to convert mg input doses to molar units (nM) for the binding equations; not reported numerically in Papachristos 2020. The 149 kDa value is the standard reference for the marketed antibody and is documented as a model assumption in the model file’s model() block.
  • Free VEGF-A effective molecular weight (MW_VEGF ≈ 45,000 g/mol). Back-calculated from the paper’s BM0 dual-unit reporting (0.0137 nM ≡ 616.5 ng/L → MW = 45 kDa, consistent with the homodimeric VEGF-A165 form). Documented as a model assumption.
  • 1:1 stoichiometric drug-target binding. The paper states “assuming a 1:1 molecular interaction” (section 2.3). The packaged QSS model uses 1:1 binding, consistent with the paper.
  • Bevacizumab observation = total drug (free + complex). The paper’s ELISA assay (section 4.4) is not described as a free-drug assay; total-drug ELISA is the typical default. Flagged as an interpretation assumption.
  • k_int = CL / V1. Per paper section 2.3: “the elimination clearance of the bevacizumab-VEGF-A complex… was set equal to the CL of free bevacizumab.” Implemented exactly as stated.
  • No errata identified. A literature search (April 2026) found no published corrigendum / erratum to Papachristos 2020.